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It is a job interview question.
So, what's the value of a vanilla European call option of infinite maturity, and a given strike, vol, interest rate, spot price.
I think, the answer should be "zero".
The contract never pays, because infinite maturity will never be reached.
It should not be equal to the spot price, which BS formula suggests in the limit T goes to infinity, I think.
If it were an American call with infinite maturity, the price could be anywhere between S and S-K.
So we are assuming that the stock does not pay any dividend? Then it should be $S_0$, which is the limiting behaviour of the BS formula. $\Phi ( d_1)$ goes to 1 as maturity approaches infinity, and $\Phi (d_2)$ can only be between 0 and 1, so the $e^{-r\tau}$ will make the $K e^{-r\tau} \Phi (d_2)$ zero.
I can see the argument around the payoff at infinity not of any value, but then if the stock never pays dividend, then can’t you make the same argument about the stock payoff? So call option price equal to $S_0$ make sense.
For a dividend paying stock, it should be zero as you can easily verify by looking at the extended BS formula. The second term goes to zero as above, but the first term now has got the term $e^{-r_f \tau}$ so it goes to zero as well. Again this makes sense because the stock value will mainly be coming from the dividend stream, which the option lacks, so its worth nothing.
I think @Gordon nails it with the comment that as the European option can never be exercised if its maturity is infinite (sloppy language) its value will be zero.
A straightforward derivation, assuming the price of a zero coupon goes to zero as maturity date goes to infinity, is as follows (I'll just do the put option, call option similar) for a potentially dividend paying stock:
$$ P = e^{-rT} E [ (K - S_T)_+] $$ By Jensen's inequality: $$ E [ (K - S_T)_+] \geq (K - E [S_T])_+ = (K - Se^{(r-q)T})_+ $$ So $$ P \geq (Ke^{-rT} - Se^{-qT})_+ $$ For the upper bound it is clear that $$ P \leq Ke^{-rT} $$ Therefore $$ (Ke^{-rT} - Se^{-qT})_+ \leq P \leq Ke^{-rT} $$ and $P \to 0$ as $T \to \infty$.
Turning this argument around you could maybe even postulate that the price of an infinite maturity zero coupon bond must be zero for otherwise you could have a nonzero valued European put option that can never be exercised, which to say the least is weird.
EDIT:
Following some exchanges with Hans (see comments below), I think I was too naive with the call option. Following the limit argument as I did above for the put will give current value of call option equal to $S_0$. I am still inclined to think that economically this does not make sense as pointed out by Gordon. I suspect it has to do with the operation of taking limits. In particular, unlike to put option payoff, the call option payoff is unbounded since $S_T$ is unbounded. So taking limits and declaring it's equal to $S_0$ may not be permissible.
And indeed, as already pointed out by Gordon in a comment, because $T \to \infty$ to begin with there is no well-defined terminal condition, and hence no well-defined solution to the PDE.
It's a great interview question by the way.
I'm not sure there has ever existed a perpetual European call option, but I'm happy to indulge in the thought process. I say it can't be worth zero, because there are certain events that cause it to have value: a) if the option were subject to variation margin according to the market value, then obviously the market could decide the value is non zero. In this situation, theoretical arguments are moot- if you have sold the option, you will have to margin it, and if the other side of the trade has more liquidity than you, you will lose the battle. b) if there is no variation margin, you have a stronger argument- for one thing, who are you buying the option from? Given an infinite time frame, they will eventually go bankrupt. But there is still a potential for value: what if the underlying stock is subject to a corporate event such as a cash acquisition? Depending on the documentation, that could deliver cash to option holders. Even if the company goes bankrupt, there is sometimes residual value for stockholders.
So I say it cannot be zero, although you have a strong case that it is nowhere near the BS formula limit of S.
This is a thought exercise, and I see two ways to think about it - one from the mathematical standpoint, in which the limit value of the black-scholes model is taken as t approaches infinity.
But the black-scholes model (and most other option models) value the option by determining a probability distribution of the payoff at expiry. But if the option is European and the maturity is infinite, then there is no payoff and thus the option is worthless. The theoretical option value based on black-scholes might be S0 but the practical value is zero since you can never extract that value.
It would be like being handed a check that you could never cash.
There is a similar thought exercise for non-dividend-paying stocks. If a company lives indefinitely and never pays a dividend, does its stock have any value? If you can never extract your ownership piece (which is what stocks represent) then the stock is worthless. It's only the expectation of liquidation at some point (via merger, acquisition, etc.) or siphoning off value through dividends that a stock has any present value.
That said, as an interview question I would be more impressed if someone discussed the pros and cons of each method rather than asserting that one was right and one was wrong, since the premise is impractical anyway. The point of these types of question is to see how you think and reason through a problem (obviously demonstrating a knowledge of the concepts along the way) more than coming up with the "right" answer.
In the standard BS model the price follows a GBM so the company can never be worth zero. But, in practice, the underlying can become worthless.
In the GBM case,the price equals $s_0$. But, in reality there is a positive probability that $S$ becomes worthless and stays that way forever. Therefore, $E[S_T]$ must tend to zero as $T$ tends to infinity and the option must be worthless. If we find an asset that truly can keep its value forever, then $s_0$ seems fair? An example: at the end of times you have the option to jump back in time to today. Even though the point will never be reached, it will still always have the same value as your life has today.
